I had a further look at the code of $specint.
The main idea of $SPECINT is to transform the special functions including log
and erf to a Hypergeometric function and use the known Laplace transform of a
Hypergeometric function F[p,q]([a1,...,ap],[b1,...,bq],z).
The function REF get the representation as a Hypergeometric function and after
some scaling and tests we try to get the Laplace transform in the routine
F19P220SIMP. There are about 16 functions implemented which try to use this
algorithm to get their Laplace transform.
The bad news is, that for more than a half of the functions implemented,
including log and erf, this algorithm don't work. The reason is that F19P220SIMP
only works for Hypergeometric functions F[p,q]([],[],t^m) with p+m-1<=q.
So it works for bessel_j with a Hypergeometric function 0F1, or Whittaker M
function with the representation 1F1. But not for log with a Hypergeometric
function 2F1 or the erf function. Here we have a transformation
1F1(1/2;3/2;-z^2), and because of the quadratic exponent the condition p+m-1<=q
is not true.
That's the reason why we don't get the known Laplace transforms for a lot of
functions.
Next steps:
1.
I have collected the formulas for the Laplace transforms of more specialized
Hypergeometric functions: the Hypergeometric 1F1 function, Hypergeometric 2F1
function, Whittaker M function, Whittaker W function and the Hypergeometric U
function. (The algorithm of the Laplace transform for the Whittaker W function
is implemented in the routine F16P217TEST. But the formula I have got looks more
general.)
2.
We can implement special routines to handle functions like log and erf to get
the Laplace transforms without using the Hypergeometric functions. That's more
the way the Maxima function $laplace do the integration.
3.
We can try to even more generalize the problem using the Meijer G functions. The
Laplace transform is known and we have a lot of possiblities to represent our
functions as Meijer G functions.
I think, the best we can do is to implement some extra code (see Point 2).
Perhaps, we can find some more general results using other known Laplace
transforms of more specialized Hypergeometric functions. The implementation of
the Meijer G functions would be the most challenging task.
A remark about the code of $specint:
To get more insight in the algorithm I have implemented some code for a noun
form of a function hypergeometric and the Laplace transform for this function.
Additionally, I have implemented the functions whittaker_m and whittaker_w. It
is much more expressive to use such nouns instead of %f, %w or %m.
Here an example with bessel_j:
(%i17) specint(%e^(-s*t)*bessel_j(n,t),t);
(%o17) s^(-n-1)/((sqrt(1/s^2+1)+1)^n*sqrt(1/s^2+1))
Now the same result using directly the Hypergeometric function:
(%i18) specint(%e^(-s*t)*(t/2)^n/gamma(n+1)*hypergeometric([],[n+1],-t^2/4),t);
(%o18) s^(-n-1)/((sqrt(1/s^2+1)+1)^n*sqrt(1/s^2+1))
The point is, I can test directly the main parts of the algorithm and don't need
to prove the implementation of the special function.
There are more functions $SPECINT uses which should be represented by expressive
noun forms, e.g.:
%l -> laguerre,
%he -> hermite,
%d -> parabolic_cylinder_d,
%h -> hankel_1 or hankel_2,
%e -> elleptic_ec
If it is interessting, I have appended a diff which shows some changes of the
code including the discussed bug fixes and extensions.
Dieter Kaiser
-----Urspr?ngliche Nachricht-----
Von: robert.dodier at gmail.com [mailto:robert.dodier at gmail.com]
Gesendet: Samstag, 17. Mai 2008 20:00
An: Dieter Kaiser
Cc: maxima at math.utexas.edu
Betreff: Re: [Maxima] Problems with the function $specint
On 5/15/08, Dieter Kaiser <drdieterkaiser at web.de> wrote:
> Because I have only time in the evening hours and I use the time to look at
the
> code, I can not answer so fast.
No problem, Dieter. Thanks a lot for looking at specint, I appreciate
your help very much. I look forward to seeing the patches when you
can post them.
Thanks again for your help,
Robert Dodier
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